Chicago is served by two major airports. O'Hare International (ORD) is one of the busiest in the world, with non-stop flights from Europe, Asia, and across North America. Midway (MDW) is smaller, mostly domestic, and often cheaper. Both connect to the city center by the L elevated train -- the Blue Line from O'Hare takes about 45 minutes and costs under $5, making it one of the best airport links in any American city. The Orange Line from Midway takes around 30 minutes.
By train, Chicago's Union Station is Amtrak's midwest hub. The California Zephyr connects to San Francisco, the Empire Builder to Seattle, and the Lake Shore Limited to New York. Regional services connect daily to Milwaukee (90 minutes), St. Louis (around 5 hours), and Detroit (around 5 hours). For trips within 3 hours, Amtrak is often more relaxed than flying once you factor in airport time.
By road, Chicago sits at the intersection of several major interstates. I-90/94 is the main north-south corridor. The drive from Milwaukee takes about 90 minutes; from Indianapolis around 3 hours; from St. Louis around 4.5 hours. Be aware that Chicago traffic on the expressways can be brutal during rush hours.
Greyhound and Megabus both serve Chicago, with particularly frequent and cheap service on the Chicago-Milwaukee and Chicago-Detroit corridors.